METHODS OF APPLIED MATHEMATICS FOR ENGINEERS AND SCIENTISTSassets.cambridge.org/97811070/04122/frontmatter/9781107004122... · METHODS OF APPLIED MATHEMATICS FOR ENGINEERS AND SCIENTISTS

METHODS OF APPLIED MATHEMATICS FORENGINEERS AND SCIENTISTS

Based on course notes from more than twenty years of teaching engi-neering and physical sciences at Michigan Technological University,Tomas B. Co’s engineering mathematics textbook is rich with examples,applications, and exercises. Professor Co uses analytical approaches tosolve smaller problems to provide mathematical insight and under-standing, and numerical methods for large and complex problems. Thebook emphasizes applying matrices with strong attention to matrixstructure and computational issues such as sparsity and efficiency.Chapters on vector calculus and integral theorems are used to buildcoordinate-free physical models, with special emphasis on orthogonalcoordinates. Chapters on ordinary differential equations and partialdifferential equations cover both analytical and numerical approaches.Topics on analytical solutions include similarity transform methods,direct formulas for series solutions, bifurcation analysis, Lagrange-Charpit formulas, and shocks/rarefaction. Topics on numerical meth-ods include stability analysis, differential algebraic equations, high-order finite-difference formulas, and Delaunay meshes. MATLABimplementations of the methods and concepts are fully integrated.

Tomas B. Co is an associate professor of chemical engineering atMichigan Technological University. After completing his PhD in chem-ical engineering at the University of Massachusetts at Amherst, he wasa postdoctoral researcher at Lehigh University, a visiting researcherat Honeywell Corp., and a visiting professor at Korea University.He has been teaching applied mathematics to graduate and advancedundergraduate students at Michigan Tech for more than twentyyears. His research areas include advanced process control, includ-ing plantwide control, nonlinear control, and fuzzy logic. His journalpublications span broad areas in such journals as IEEE Transactionsin Automatic Control, Automatica, AIChE Journal, Computers inChemical Engineering, and Chemical Engineering Progress. He hasbeen nominated twice for the Distinguished Teaching Award atMichigan Tech and is a member of the Michigan TechnologicalUniversity Academy of Teaching Excellence.

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Cambridge University Press978-1-107-00412-2 - Methods of Applied Mathematics for Engineers and ScientistsTomas B. CoFrontmatterMore information

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Methods of Applied Mathematics forEngineers and Scientists

Tomas B. CoMichigan Technological University






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© Tomas B. Co 2013

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Co, Tomas B., 1959–Methods of applied mathematics for engineers and scientists / Tomas B. Co.,Michigan Technological University.pages cmIncludes bibliographical references and index.ISBN 978-1-107-00412-2 (hardback)1. Matrices. 2. Differential equations – Numerical solutions. I. Title.QA188.C63 2013512.9′434–dc23 2012043979

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Contents

Preface page xi

I MATRIX THEORY

1 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Definitions and Notations 41.2 Fundamental Matrix Operations 61.3 Properties of Matrix Operations 181.4 Block Matrix Operations 301.5 Matrix Calculus 311.6 Sparse Matrices 391.7 Exercises 41

2 Solution of Multiple Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.1 Gauss-Jordan Elimination 552.2 LU Decomposition 592.3 Direct Matrix Splitting 652.4 Iterative Solution Methods 662.5 Least-Squares Solution 712.6 QR Decomposition 772.7 Conjugate Gradient Method 782.8 GMRES 792.9 Newton’s Method 802.10 Enhanced Newton Methods via Line Search 822.11 Exercises 86

3 Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.1 Matrix Operators 1003.2 Eigenvalues and Eigenvectors 1073.3 Properties of Eigenvalues and Eigenvectors 1133.4 Schur Triangularization and Normal Matrices 1163.5 Diagonalization 1173.6 Jordan Canonical Form 1183.7 Functions of Square Matrices 120

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3.8 Stability of Matrix Operators 1243.9 Singular Value Decomposition 1273.10 Polar Decomposition 1323.11 Matrix Norms 1353.12 Exercises 138

II VECTORS AND TENSORS

4 Vector and Tensor Algebra and Calculus . . . . . . . . . . . . . . . . . . . . 149

4.1 Notations and Fundamental Operations 1504.2 Vector Algebra Based on Orthonormal Basis Vectors 1544.3 Tensor Algebra 1574.4 Matrix Representation of Vectors and Tensors 1624.5 Differential Operations for Vector Functions of One Variable 1644.6 Application to Position Vectors 1654.7 Differential Operations for Vector Fields 1694.8 Curvilinear Coordinate System: Cylindrical and Spherical 1844.9 Orthogonal Curvilinear Coordinates 1894.10 Exercises 196

5 Vector Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.1 Green’s Lemma 2055.2 Divergence Theorem 2085.3 Stokes’ Theorem and Path Independence 2105.4 Applications 2155.5 Leibnitz Derivative Formula 2245.6 Exercises 225

III ORDINARY DIFFERENTIAL EQUATIONS

6 Analytical Solutions of Ordinary Differential Equations . . . . . . . . . . 235

6.1 First-Order Ordinary Differential Equations 2366.2 Separable Forms via Similarity Transformations 2386.3 Exact Differential Equations via Integrating Factors 2426.4 Second-Order Ordinary Differential Equations 2456.5 Multiple Differential Equations 2506.6 Decoupled System Descriptions via Diagonalization 2586.7 Laplace Transform Methods 2626.8 Exercises 263

7 Numerical Solution of Initial and Boundary Value Problems . . . . . . . 273

7.1 Euler Methods 2747.2 Runge Kutta Methods 2767.3 Multistep Methods 2827.4 Difference Equations and Stability 2917.5 Boundary Value Problems 2997.6 Differential Algebraic Equations 3037.7 Exercises 305






Contents vii

8 Qualitative Analysis of Ordinary Differential Equations . . . . . . . . . . 311

8.1 Existence and Uniqueness 3128.2 Autonomous Systems and Equilibrium Points 3138.3 Integral Curves, Phase Space, Flows, and Trajectories 3148.4 Lyapunov and Asymptotic Stability 3178.5 Phase-Plane Analysis of Linear Second-Order

Autonomous Systems 3218.6 Linearization Around Equilibrium Points 3278.7 Method of Lyapunov Functions 3308.8 Limit Cycles 3328.9 Bifurcation Analysis 3408.10 Exercises 340

9 Series Solutions of Linear Ordinary Differential Equations . . . . . . . . 347

9.1 Power Series Solutions 3479.2 Legendre Equations 3589.3 Bessel Equations 3639.4 Properties and Identities of Bessel Functions and

Modified Bessel Functions 3699.5 Exercises 371

IV PARTIAL DIFFERENTIAL EQUATIONS

10 First-Order Partial Differential Equations and the Method ofCharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

10.1 The Method of Characteristics 38010.2 Alternate Forms and General Solutions 38710.3 The Lagrange-Charpit Method 38910.4 Classification Based on Principal Parts 39310.5 Hyperbolic Systems of Equations 39710.6 Exercises 399

11 Linear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . 405

11.1 Linear Partial Differential Operator 40611.2 Reducible Linear Partial Differential Equations 40811.3 Method of Separation of Variables 41111.4 Nonhomogeneous Partial Differential Equations 43111.5 Similarity Transformations 43911.6 Exercises 443

12 Integral Transform Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

12.1 General Integral Transforms 45112.2 Fourier Transforms 45212.3 Solution of PDEs Using Fourier Transforms 45912.4 Laplace Transforms 46412.5 Solution of PDEs Using Laplace Transforms 47412.6 Method of Images 47612.7 Exercises 477






viii Contents

13 Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

13.1 Finite Difference Approximations 48413.2 Time-Independent Equations 49113.3 Time-Dependent Equations 50413.4 Stability Analysis 51213.5 Exercises 519

14 Method of Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

14.1 The Weak Form 52414.2 Triangular Finite Elements 52714.3 Assembly of Finite Elements 53314.4 Mesh Generation 53914.5 Summary of Finite Element Method 54114.6 Axisymmetric Case 54614.7 Time-Dependent Systems 54714.8 Exercises 552

Bibliography B-1

Index I-1

A Additional Details and Fortification for Chapter 1 . . . . . . . . . . . . . . 561

A.1 Matrix Classes and Special Matrices 561A.2 Motivation for Matrix Operations from Solution of Equations 568A.3 Taylor Series Expansion 572A.4 Proofs for Lemma and Theorems of Chapter 1 576A.5 Positive Definite Matrices 586

B Additional Details and Fortification for Chapter 2 . . . . . . . . . . . . . 589

B.1 Gauss Jordan Elimination Algorithm 589B.2 SVD to Determine Gauss-Jordan Matrices Q and W 594B.3 Boolean Matrices and Reducible Matrices 595B.4 Reduction of Matrix Bandwidth 600B.5 Block LU Decomposition 602B.6 Matrix Splitting: Diakoptic Method and Schur

Complement Method 605B.7 Linear Vector Algebra: Fundamental Concepts 611B.8 Determination of Linear Independence of Functions 614B.9 Gram-Schmidt Orthogonalization 616B.10 Proofs for Lemma and Theorems in Chapter 2 617B.11 Conjugate Gradient Algorithm 620B.12 GMRES Algorithm 629B.13 Enhanced-Newton Using Double-Dogleg Method 635B.14 Nonlinear Least Squares via Levenberg-Marquardt 639

C Additional Details and Fortification for Chapter 3 . . . . . . . . . . . . . . 644

C.1 Proofs of Lemmas and Theorems of Chapter 3 644C.2 QR Method for Eigenvalue Calculations 649C.3 Calculations for the Jordan Decomposition 655






Contents ix

C.4 Schur Triangularization and SVD 658C.5 Sylvester’s Matrix Theorem 659C.6 Danilevskii Method for Characteristic Polynomial 660

D Additional Details and Fortification for Chapter 4 . . . . . . . . . . . . . . 664

D.1 Proofs of Identities of Differential Operators 664D.2 Derivation of Formulas in Cylindrical Coordinates 666D.3 Derivation of Formulas in Spherical Coordinates 669

E Additional Details and Fortification for Chapter 5 . . . . . . . . . . . . . . 673

E.1 Line Integrals 673E.2 Surface Integrals 678E.3 Volume Integrals 684E.4 Gauss-Legendre Quadrature 687E.5 Proofs of Integral Theorems 691

F Additional Details and Fortification for Chapter 6 . . . . . . . . . . . . . . 700

F.1 Supplemental Methods for Solving First-Order ODEs 700F.2 Singular Solutions 703F.3 Finite Series Solution of dx/dt = Ax + b(t) 705F.4 Proof for Lemmas and Theorems in Chapter 6 708

G Additional Details and Fortification for Chapter 7 . . . . . . . . . . . . . . 715

G.1 Differential Equation Solvers in MATLAB 715G.2 Derivation of Fourth-Order Runge Kutta Method 718G.3 Adams-Bashforth Parameters 722G.4 Variable Step Sizes for BDF 723G.5 Error Control by Varying Step Size 724G.6 Proof of Solution of Difference Equation, Theorem 7.1 730G.7 Nonlinear Boundary Value Problems 731G.8 Ricatti Equation Method 734

H Additional Details and Fortification for Chapter 8 . . . . . . . . . . . . . . 738

H.1 Bifurcation Analysis 738

I Additional Details and Fortification for Chapter 9 . . . . . . . . . . . . . . 745

I.1 Details on Series Solution of Second-Order Systems 745I.2 Method of Order Reduction 748I.3 Examples of Solution of Regular Singular Points 750I.4 Series Solution of Legendre Equations 753I.5 Series Solution of Bessel Equations 757I.6 Proofs for Lemmas and Theorems in Chapter 9 761

J Additional Details and Fortification for Chapter 10 . . . . . . . . . . . . . 771

J.1 Shocks and Rarefaction 771J.2 Classification of Second-Order Semilinear Equations: n > 2 781J.3 Classification of High-Order Semilinear Equations 784






x Contents

K Additional Details and Fortification for Chapter 11 . . . . . . . . . . . . . 786

K.1 d’Alembert Solutions 786K.2 Proofs of Lemmas and Theorems in Chapter 11 791

L Additional Details and Fortification for Chapter 12 . . . . . . . . . . . . . 795

L.1 The Fast Fourier Transform 795L.2 Integration of Complex Functions 799L.3 Dirichlet Conditions and the Fourier Integral Theorem 819L.4 Brief Introduction to Distribution Theory and Delta Distributions 820L.5 Tempered Distributions and Fourier Transforms 830L.6 Supplemental Lemmas, Theorems, and Proofs 836L.7 More Examples of Laplace Transform Solutions 840L.8 Proofs of Theorems Used in Distribution Theory 846

M Additional Details and Fortification for Chapter 13 . . . . . . . . . . . . . 851

M.1 Method of Undetermined Coefficients for FiniteDifference Approximation of Mixed Partial Derivative 851

M.2 Finite Difference Formulas for 3D Cases 852M.3 Finite Difference Solutions of Linear Hyperbolic Equations 855M.4 Alternating Direction Implicit (ADI) Schemes 863

N Additional Details and Fortification for Chapter 14 . . . . . . . . . . . . . 867

N.1 Convex Hull Algorithm 867N.2 Stabilization via Streamline-Upwind Petrov-Galerkin (SUPG) 870






Preface

This book was written as a textbook on applied mathematics for engineers and sci-entist, with the expressed goal of merging both analytical and numerical methodsmore tightly than other textbooks. The role of applied mathematics has continued togrow increasingly important with advancement of science and technology, rangingfrom modeling and analysis of natural phenomenon to simulation and optimizationof man-made systems. With the huge and rapid advances of computing technol-ogy, larger and more complex problems can now be tackled and analyzed in a verytimely fashion. In several cases, what used to require supercomputers can now besolved using personal computers. Nonetheless, as the technological tools continueto progress, it has become even more imperative that the results can be understoodand interpreted clearly and correctly, as well as the need for a deeper knowledgebehind the strengths and limitations of the numerical methods used. This meansthat we cannot forgo the analytical techniques because they continue to provideindispensable insights on the veracity and meaning of the results. The analyticaltools continue to be of prime importance for basic understanding for building math-ematical models and data analysis. Still, when it comes to solving large and complexproblems, numerical methods are needed.

The level of exposition in this book is aimed at graduate students, advancedundergraduate students, and researchers in the engineering and science field. Thusthe topics were mostly chosen to continue several topics found in most undergradu-ate textbooks in applied mathematics. We have focused on advanced concepts andimplementation of various mathematical tools to solve the problems that most grad-uate students are likely to face in their research work and other advanced courses.

The contents of the book can be divided into four main parts: matrix theory,vectors and tensors, ordinary differential equations, and partial differential equa-tions. We begin the book with matrix theory because the tools developed in matrixtheory form the crucial foundations used in the rest of the book. The next part cen-ters on the concepts used in vector and tensor theory, including the application oftensor calculus and integral theorems to develop mathematical models of physicalsystems, often resulting in several differential equations. The last two parts focus onthe solution of ordinary and partial differential equations. It can be argued that theprimary needs of applied mathematics in engineering and the physical sciences areto obtain models for a system or phenomena in the form of differential equations

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xii Preface

and then to be able to “solve” them to predict and understand the effects of changesin model parameters, boundary conditions, or initial conditions.

Although the methods of applied mathematics are independent of computingplatform and programs, we have chosen to use MATLAB as a particular platformunder which we investigate the mathematical methods, techniques, and ideas so thatthe approaches can be tested and the results can be visualized. The supplied MAT-LAB codes are all included on the book’s website, and the reader can modify thecodes for their own use. There are several excellent MATLAB toolboxes supplied bythird-party software developers, and they have been optimized for speed, efficiency,and user-friendliness. However, the unintended consequences of user-friendly toolscan sometimes render the users to be “button pushers.” We contend that students inapplied mathematics still need to discover the mechanism and ideas behind the full-blown programs – at least to apply them to simple test problems and gain some basicunderstanding of the various approaches. The links to the supplemental MATLABprograms and files can be accessed through the link: www.cambridge.org/Co.

The appendices are collected as chapter fortifications. They include proofs,advanced topics, additional tables, and examples. The reader should be able toaccess these materials through the web via the link: www.cambridge.org/Co.The index also contains topics that can be found in the appendices, and they aregiven page numbers that continue the count from the main text.

Several colleagues and students have helped tremendously in the writing of thistextbook. Mostly, I want to thank my best friend and wife, Faith Morrison, for thesupport and encouragement and the sacrifice she’s made so that I could finish thisextended and personally significant project. I hope the textbook will contain usefulinformation for the readers, enough for them to share in the continued exploration ofthe methods and applications of mathematics to further improve the understandingand conditions of our world.

Tomas B. CoHoughton, MI






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